College Algebra Exam Review 64

College Algebra Exam Review 64 - 74 1 ALGEBRAIC THEMES...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
74 1. ALGEBRAIC THEMES group of rotations of the square card. Show that the bijection e $ 1 r $ i r 2 $ ± 1 r 3 $ ± i produces a matching of the multiplication tables of the two groups. That is, if we apply the bijection to each entry of the multiplication table of H , we produce the multiplication table of R . Thus, the two groups are isomorphic. 1.10.4. Show that the group C 4 D f i; ± 1; ± i;1 g of fourth roots of unity in the complex numbers is isomorphic to Z 4 . The next several exercises give examples of groups coming from var- ious areas of mathematics and require some topology or real and complex analysis. Skip the exercises for which you do not have the appropriate background. 1.10.5. An isometry of R 3 is a bijective map T W R 3 ! R 3 satisfying d.T.x/;T.y// D d.x;y/ for all x;y 2 R 3 . Show that the set of isome- tries of R 3 forms a group. (You can replace R 3 with any metric space .) 1.10.6. A homeomorphism of R 3 is a bijective map T W R 3 ! R 3 such that both T and its inverse are continuous. Show that the set of homeomor-
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Ask a homework question - tutors are online